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Yes and no.

It depends on your definition of square root. By the actual one, yes. All non-negative numbers have a square root. That square root might be irrational but it has a square root, nonetheless. 10 isn't a square number because there's no integer that can be squared to make ten but 10 definitely has a square root: 3.16227766.......

If by square root you mean an integer square root, then no. If a number has an integer as its square root then you could square that integer to get the number, making it a square number.

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Q: Can numbers that are not square have square roots?
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